In this work, we present a novel method to use physics-informed neural networks in complex geometries by representing manifolds by the eigenfunctions of the Laplace-Beltrami operator. Given a mesh representing the geometry, we can compute these functions with standard finite elements. We show that this method outperforms the original formulation of PINNs, especially in cases where there points in the input domain are close in Eucledian distance but far in the intrinsic distance. We test our methodology by solving an inverse problem with the Eikonal equation and the heat equation, a forward Poisson problem, and an operator learning problem for inferring the geodesic distance between two points on a complex manifold. The Laplace-Beltrami eigenfunctions have been used in other applications of machine learning, such as point signatures in shapes [1]}, [2]}, shape descriptors [3]} and to approximate kernels on manifolds for Gaussian process regression [4]} and classification [5]}. To our knowledge this is the first time they have been used in the context of physics-informed neural networks.
